The Ulam Spiral of Primes
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scienceblogs.com
I made canvas renderer to check it:
http://alteredqualia.com/visualization/ulam-spiral.html
It's not checkerboard, but there are horizontal and vertical patterns.
let P(n) be the nth prime number: P(1)=2, etc.
let n be an integer >= 2, and m be an integer > n
then P(m) ≡ 1 and (P(n) - 1) (mod P(n))
?This construction was first made by Polish-American mathematician Stanislaw Ulam (1909-1986) in 1963 while doodling during a boring talk at a scientific meeting.
Awesome. My new excuse for all those doodles in my engineering notebook: valuable, valuable mathematical research.
Amen to that.
http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmet...
I agree that some mathematical objects are "more fundamental than others"; primes are probably among the most basic.
If you're interested in the ontological status of mathematical entities, and the process and meaning of mathematics, check out "The Mathematical Experience" by Davis and Hersh,
http://books.google.com/books?id=lMdz84dWNnAC&printsec=f...
There is a section on primes starting on page 209.
There are a number of practical social situations where an understanding of divisibility can be useful.
For instance, say a group of social animals of equal rank stumble upon some berries and split them amongst themselves. It would be beneficial for them to understand that if the number of berries is relatively prime to the number of them, then an even distribution is impossible and they should accept that.
I'm sure more compelling examples of grouping in social animals arise all the time. Now if these animals begin to study numbers abstractly, one of the first things that they will practically need to understand is the rules that govern groupings. This will lead them immediately into the idea of prime numbers (and by extension relatively prime numbers).
I can't really think of a technique of grouping or looking at numbers that wouldn't at least implicitly require prime numbers.
It may be that higher level analysis of primes (like in this article) is somewhat artificial, but I'd maintain that the idea that a number is prime, or at least the concept that two numbers are relatively prime, is one that is fundemental to having any understanding of a number system.
Your use of the word "random" seems to be a non sequitur. Could you explain that connection?
I used the word random because the distribution of Primes is random as far as we currently know. http://en.wikipedia.org/wiki/Prime_Numbers#Distribution
So no, "random as far as we currently know" is a pretty terrible description of the distribution of prime numbers.
We can prove the prime number theorem, we know roughly how many primes there are in a span of numbers and we know the approximate frequency of prime numbers in that span. We know on average how far apart the primes are in a particular span of numbers. All from the prime number theorem and related theorems. This stuff has been proven. Likewise, if the Riemann hypothesis is true then primes would have a very regular distribution.
What we can't yet do is create a generator function that catches most primes. We can create generators though (For example: (n! +1) is prime but it misses tons and tons of primes.) There are no hard fast rules that catch a majority of only primes. This probably doesn't qualify as "random distribution" unless you take a really simple definition of random. It's the non-randomness of their distribution that allows us to test the primality of numbers for cryptography.
n!+1 is not always prime. For instance, 4!+1 = 25 = 5^2, 5!+1 = 121 = 11^2, 6!+1 = 721 = 7.103.
If RH is true, then the number of primes < x differs from the naive prediction you get from saying n is prime with probability 1/log(n) by at most something like sqrt(x) log(x). That's entirely comparable to the accuracy you'd get if the primes really were randomly distributed. (More precisely: the variance of something that's 1 with probability p, else 0, is p(1-p), which is roughly equal to p when p is small, and variances of independent things are additive, so on the random-distribution hypothesis the variance of pi(x) would be comparable to the sum of 1/log(n), which is to say n/log(n). I haven't actually done the calculations, but I think this means that with probability 1 some bound of the same type that you get from RH should hold. So the bound you get from RH doesn't show that the primes are any more regularly distributed than the simple random model suggests.
Similarly, "primes" generated according to the simple random model would, with probability 1, obey theorems of the sort that we know how to prove about the distribution of primes. (And some that we don't know how to prove.)
It is possible to define mathematical functions that recognize primes, or that take on all the primes (and nothing else) as values. But they're generally contrived and inefficient to evaluate.
The closest I can come is the coincidence that a random variable is used in the proof of the Prime Number Theorem, but that's just a modeling technique.
At least I did find this awesome quotation, which seems to play both sides:
In a 1975 lecture, D. Zagier commented "There are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition and role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, and nobody can predict where the next one will sprout. The second fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behavior, and that they obey these laws with almost military precision" (Havil 2003, p. 171).
I confess that I only have an undergraduate degree in Mathematics, but with that caveat I have a very different understanding: importance is a value judgement, and concepts in number theory such as prime-ness hinge on mere observations about the properties of numbers. People may assign importance to the property, but the property exists independently of how people feel about it.
Once you have a system of numbering, one of the first things one does is to see which numbers can be arrived at by simply adding other numbers to themselves a bunch of times.
I have a pretty limited knowledge of number theory, so I may be completely off mark, but I don't know that people have yet arrived at a nice fundamental theory that can encapsulate the way that primes arrange themselves along the number line.